L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (2 + 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s − 6·13-s − 0.999·15-s + (2.5 − 4.33i)17-s + (0.5 + 0.866i)19-s + (−2.5 + 0.866i)21-s + (−3.5 − 6.06i)23-s + (2 − 3.46i)25-s − 5·27-s + 2·29-s + (−2.5 + 4.33i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.755 + 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s − 1.66·13-s − 0.258·15-s + (0.606 − 1.05i)17-s + (0.114 + 0.198i)19-s + (−0.545 + 0.188i)21-s + (−0.729 − 1.26i)23-s + (0.400 − 0.692i)25-s − 0.962·27-s + 0.371·29-s + (−0.449 + 0.777i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.953206 + 0.399472i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.953206 + 0.399472i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-2.5 - 4.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.26128449298071632015483819022, −12.49513974671558037969982310582, −11.65692809887798795583277751402, −10.57555685519585205276805160244, −9.698304938160205601973786242431, −8.379656881043203736225585759727, −7.14043922179287005866057068800, −5.57360598038331244081150557594, −4.59725453899486130670035939670, −2.52824303702105238294123305358,
1.60478291472805514387162796273, 4.11761957989625921924448635341, 5.44255037315216003872634593352, 7.00594909735142233711115444878, 7.74974602213413377460865354291, 9.376302927091470684983633768693, 10.23119771454001353389512992432, 11.73197044916876882571546769312, 12.36267530916875766472585257593, 13.36637103897624603813564426192